On the maximum field of linearity of linear sets

Let (Formula presented.) denote an (Formula presented.) -dimensional (Formula presented.) -vector space. For an (Formula presented.) -dimensional (Formula presented.) -subspace (Formula presented.) of (Formula presented.), assume that (Formula presented.) for each nonzero vector (Formula presented.). If (Formula presented.), then we prove the existence of an integer (Formula presented.) such that the set of one-dimensional (Formula presented.) -subspaces generated by nonzero vectors of (Formula presented.) is the same as the set of one-dimensional (Formula presented.) -subspaces generated by nonzero vectors of (Formula presented.). If we view (Formula presented.) as a point set of (Formula presented.), it means that (Formula presented.) and (Formula presented.) determine the same set of directions. We prove a stronger statement when (Formula presented.). In terms of linear sets, it means that an (Formula presented.) -linear set of (Formula presented.) has maximum field of linearity (Formula presented.) only if it has a point of weight one. We also present some consequences regarding the size of a linear set.